METODY ILOŚCIOWE W BADANIACH EKONOMICZNYCH

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1 Warsaw Unversty of Lfe Scences SGGW Faculty of Appled Informatcs and Mathematcs METODY ILOŚCIOWE W BADANIACH EKONOMICZNYCH QUANTITATIVE METHODS IN ECONOMICS Volume XII, No. EDITOR-IN-CHIEF Bolesław Borkowsk Warsaw 20

2 EDITORIAL BOARD Prof. Zbgnew Bnderman char, Prof. Bolesław Borkowsk, Prof. Leszek Kuchar, Prof. Wojcech Zelńsk, Dr. hab. Stansław Gędek, Dr. Hanna Dudek, Dr. Agata Bnderman Secretary SCIENTIFIC BOARD prof. Bolesław Borkowsk char (Warsaw Unversty of Lfe Scences SGGW), prof. Zbgnew Bnderman (Warsaw Unversty of Lfe Scences SGGW), prof. Paolo Gajo (Unversty of Florence, Italy), prof. Evgeny Grebenkov (Computng Centre of Russa Academy of Scences, Moscow, Russa), prof. Yury Kondratenko (Black Sea State Unversty, Ukrane), prof. Vassls Kostoglou (Alexander Technologcal Educatonal Insttute of Thessalonk, Greece), prof. Robert Kragler (Unversty of Appled Scences, Wengarten, Germany), prof. Yochanan Shachmurove (The Cty College of The Cty Unversty of New York), prof. Alexander N. Prokopenya (Brest Unversty, Belarus), Dr. Monka Krawec Secretary (Warsaw Unversty of Lfe Scences SGGW). PREPARATION OF THE CAMERA READY COPY Dr. Jolanta Kotlarska, Dr. Elżbeta Saganowska TECHNICAL EDITORS Dr. Jolanta Kotlarska, Dr. Elżbeta Saganowska LIST OF REVIEWERS Prof. Iacopo Bernett (Unversty of Florence, Italy) Prof. Paolo Gajo (Unversty of Florence, Italy) Prof. Yury Kondratenko (Black Sea State Unversty, Ukrane) Prof. Vassls Kostoglou (Alexander Technologcal Educatonal Insttute of Thessalonk, Greece), Prof. Karol Kukuła (Unversty of Agrculture n Krakow) Prof. Wanda Marcnkowska Lewandowska (Warsaw School of Economcs) Prof. Yochanan Shachmurove (The Cty College of the Cty Unversty of New York) Prof. Ewa Marta Syczewska (Warsaw School of Economcs) Prof. Dorota Wtkowska (Warsaw Unversty of Lfe Scences SGGW) Prof. Wojcech Zelńsk (Warsaw Unversty of Lfe Scences SGGW) Dr. Lucyna Błażejczyk Majka (Adam Mckewcz Unversty n Poznan) Dr. Mchaela Chocolata (Unversty of Economcs n Bratslava, Slovaka) ISSN X Copyrght by Katedra Ekonometr Statystyk SGGW Warsaw 20, Volume XII, No. The orgnal verson s the paper verson Publshed by Warsaw Unversty of Lfe Scences Press

3 CONTENTS Zbgnew Bnderman, Bolesław Borkowsk, Wesław Szczesny An applcaton of radar charts to geometrcal measures of structures of conformablty... Zbgnew Bnderman, Marek Werzbck Some remarks on aplcatons of algebrac analyss to economcs... 5 Mchaela Chocholatá Tradng volume and volatlty of stock returns: Evdence from some European and Asan stock markets Marcn Dudzńsk, Konrad Furmańczyk The quantle estmaton of the maxma of sea levels E. M. Ekanayake, Lucyna Korneck Factors affectng nward foregn drect nvestment flows nto the Unted States: Evdence from State-Level Data Andrea Furková, Kvetoslava Surmanová Stochastc fronter analyss of regonal compettveness Jarosław Jankowsk Identfcaton of web platforms usage patterns wth dynamc tme seres analyss methods Stansław Jaworsk, Konrad Furmańczyk On the choce of parameters of change-pont detecton wth applcaton to stock exchange data Krzysztof Karpo, Arkadusz Orłowsk, Potr Łukasewcz, Jerzy Różańsk Some applcatons of rank clocks method Monka Krawec Effcency of ndrect ways of nvestng n commodtes n condtons of polsh captal market Mara Parlńska, Galsan Dareev Applcatons of producton functon n agrculture... 9 Grzegorz Przekota, Tadeusz Waścńsk, Lda Sobczak Reacton of the nterest rates n Poland to the nterest rates changes n the USA and euro zone... 25

4 Contents Aleksander Strasburger, Andrzej Zembrzusk On applcaton of Newton s Method to solve optmzaton problems n the consumer theory. Expanson s Paths and Engel Curves Ewa Marta Syczewska Contegraton snce Granger: evoluton and development Tadeusz Waścńsk, Grzegorz Przekota, Lda Sobczak Behavor of the Central Europe exchange rates to the Euro and US dollar Wojcech Zelńsk Comparson of confdence ntervals for fracton n fnte populatons... 77

5 QUANTITATIVE METHODS IN ECONOMICS Vol. XII, No., 20, pp. 3 AN APPLICATION OF RADAR CHARTS TO GEOMETRICAL MEASURES OF STRUCTURES OF CONFORMABILITY Zbgnew Bnderman, Bolesław Borkowsk Department of Econometrcs and Statstcs, Warsaw Unversty of Lve Scences SGGW e-mals: zbgnew_bnderman@sggw.pl; boleslaw_borkowsk@sggw.pl Wesław Szczesny Department of Informatcs, Warsaw Unversty of Lve Scences SGGW e-mal: weslaw_szczesny@sggw.pl Abstract: In the followng work we presented a method of usng radar charts to calculate measures of conformablty of two objects accordng to formulas gven by, among others, Dce, Jaccard, Tanmoto and Tversky. Ths method ncorporates another one presented by the authors of ths study [Bnderman, Borkowsk, Szczesny 200]. Presented methods can be also utlzed to defne smlartes between gven objects, as well as to order and group objects. Measures descrbed n ths work satsfy the condton of stablty as they do not depend on the order of studed features. Key words: radar method, radar measure of conformablty, Dce s, Jaccard s measure of smlarty, synthetc measures, classfcaton, cluster analyss. CONSTRUCTION OF RADAR MEASURES OF CONFORMABILITY In prevous works authors used methods that have a smple nterpretaton n the form of a radar chart to order, classfy and measure smlarty of objects [Bnderman, Borkowsk, Szczesny 2008, 2009, 2009a, 200, 200a, b, c, d, Bnderman, Szczesny 2009, 20, Bnderman 2009, 2009a]. Those methods do not depend on the way the features of a gven object are ordered. In the followng work authors attempted to utlze those methods n other, wdely known means of measurng smlarty between two objects. Comparng structures of objects s chosen here as an example. Coeffcents of Jaccard, Dce and Tanmoto, Tversky ndex and cosne smlarty are all exemplary geometrcal measures of smlarty.

6 2 Zbgnew Bnderman, Bolesław Borkowsk, Wesław Szczesny The methods presented here may seem numercally complcated but n the age of computers ths problem s of lttle sgnfcance. Numerous studes conducted n many dfferent felds of scence: economcs, statstcs, computer scence, chemstry, bology, ecology, psychology, culture and toursm have proven the usefulness of those methods [Bnderman 2009a, Bnderman, Borkowsk, Szczesny 200b, c, Cok, Kowalczyk, Pleszczyńska, Szczesny 995, Deza E., Deza M.M. 2006, Duda, Hart, Stork 2000, Gordon 999, Hubalek 982, Kukuła 2000, 200, Legendre P., Legendre L. 998, Monev 2004, Szczesny 2002, Tan, Stenbach, Kumar 2006, Warrens 2008]. Let Q and P be two objects that are descrbed by a set of n (n>2) features. n Assume that objects Q and P are descrbed by two vectors x, y R +, where: x = ( x, x2,..., xn), y = ( y, y2,..., yn); x, y 0; = 2,,..., n and n n x =, y =. = = In order to graphcally represent the methods we nscrbe a regular n-gon nto a unt crcle (wth a radus of ) wth a centre n the orgn of a polar coordnate system 0uv and we wll connect the vertces of ths n-gon wth the orgn of the coordnate system. Thus, one constructs lne segments of length, we wll denote, n sequence, O,O2,...,0n, startng, for defntveness, wth the lne segment coverng w axs. Assume that at least two coordnates of each of the vectors x and y are non-zero. Because features of objects x and y take on values from an nterval <0,>, that s 0 x 0 x, 0 y 0 y, =,2,...n, where 0:=(0,0,...,0), :=(,,...,), we can represent the values of those features as a radar chart. To do so, let x (y ) denote those ponts on the 0 axs that came nto beng by ntersectng the 0 axs wth a crcle wth the centre at the orgn of the coordnate system and radus of x (y ), =,2,...,n. By connectng the ponts: x wth x 2, x 2 wth x 3,..., x n wth x (y wth y 2, y 2 wth y 3,..., y n wth y ) we get n-gons S Q and S P, where ts areas S Q and S P, are gven by formulas: n n 2π 2π SQ = Sx = : n 2xx sn = sn, gdze, + n 2 n xx x = + + x = = () n n 2π 2π SP = Sy = sn sn, gdze :. 2yy = n 2 yy y = y + n + n+ = = The formula for the area of the ntersecton of those n-gons, whch we wll denote by Sx y : = Sx Sy has a more complcated form. Its form and detaled determnaton can be found n [Bnderman, Borkowsk, Szczesny 200]. Usng those formulae we can denote the area of the unon of n-gons S x and S y as

7 An applcaton of radar charts 3 Sx Sy = Sx + Sy Sx S y, (2) where the areas Sx, Sy are defned by formulae (). Fgure presents two graphcal llustratons of vectors x=(0,2, 0,2, 0,3, 0,5, 0,, 0,05) and y =(0,5, 0,5, 0,2, 0,25, 0,5, 0,) that descrbe two exemplary demographcal structures (for age ranges: 0-4, 5-24, 25-49, 50-64, 65-79, >80), whle Fg. A and B dffer only by the order of axes (meanng the permutaton of the coordnates). Fg.. Radar charts for vectors x and y, whch coordnates present two exemplary demographcal structures, by dfferent orderng of axes. > ,3 0,25 0,2 0,5 0, 0, > B 0-4 0,3 0,25 0,2 0,5 0, 0, A Source: own work From the fgure t s clear that areas of n-gons Sx, S y and ther unons on fgures A and B dffer n sze. They are: 0,076; 0,074; 0,05 and 0,075; 0,069; 0,047, respectvely. In works [Bnderman Borkowsk, Szczesny 2008, 200] authors proposed a measure of conformablty of objects that uses a geometrcal nterpretaton n the form of radar charts and s defned as follows: R xy = S S x y σ xy S S x y ω xy for n= 3 for n 4, () 3

8 4 Zbgnew Bnderman, Bolesław Borkowsk, Wesław Szczesny where mn( S gdy > 0 gdy > 0 x, S y ) S x S y max( S x, S y ) S x S y xy gdy Sx Sy= 0 gdy Sx Sy= 0 σ : =, ω : = xy. Note that such a measure of conformablty (smlarty) has the property of: 0 μ x.y and depends on the orderng of features [cf. Bnderman Borkowsk, Szczesny 2008]. To defne a measure of conformablty of objects that does not depend on the orderng of features, let us denote by π j a j-th permutaton of numbers,2,,n. It s known that the number of all such permutatons s equal to n! [Mostowsk, Stark 977]. Each permutaton of coordnates of vectors x and y corresponds to one permutaton π j. Let vectors x j, y j denote the j-th permutaton of coordnates of vectors x and y, respectvely, assumng that x :=x, y :=y. For example, f n=3, x=(x,x 2,x 3 ), y=(y,y 2,y 3 ) and π =(,2,3), π 2 =(,3,2), π 3 =(2,,3), π 4 =(2,3,), π 5 =(3,,2), π 6 =(3,2,) then: x =(x,x 2,x 3 ), y =(y,y 2,y 3 ), x 2 =(x,x 3,x 2 ), y 2 =(y,y 3,y 2 ) x 3 =(x 2,x,x 3 ), y 3 =(y 2,y,y 3 ), x 4 =(x 2,x 3,x ), y 4 =(y 2,y 3,y ), x 5 =(x 3,x,x 2 ), y 5 =(y 3,y,y 2 ), x 6 =(x 3,x 2,x ), y 6 =(y 3,y 2,y ). A result from our earler works s that a coeffcent of conformablty of structures corresponds to each j-th permutaton x j, y j of coordnates of vectors x and y j RQP, = R xy, ( 4) j j where naturally RQP, = R xy. Therefore, we can assume that the followng desgnatons of three dfferent measures of conformablty of consdered objects Q and P. Naturally, those measures are nvarant under the orderng of coordnates for vectors x and y. M M j QP, = xy = QP, j n! m m j QP, = xy = QP, j n! n! s s j QP, = xy = QP,. n! j= R R max R, R R mn R, () 5 R R R

9 An applcaton of radar charts 5 Other well-known n lterature technques that use geometrcal nterpretatons, such as radar charts, may be used to compare two structures x = ( x, x2,..., xn), y = ( y, y2,..., y n). Most well known among them are: cosne smlarty [Deza, Deza 2006] Sx Sy for S S > 0 c x y S S, xy x y (6) = 0 for Sx Sy = 0 Jaccard coeffcent [Jaccard 90, 902, 908] Sx Sy for S S 0 J x y > xy = Sx Sy, 0 for Sx Sy = 0 (7) Dce s coeffcent [Dce 945] 2Sx Sy for Sx Sy > 0 D xy = Sx + Sy, (8) 0 for Sx Sy = 0 Tanmoto coeffcent [Tanmoto 957, 959] Sx Sy for Sx Sy > 0 T xy = Sx + Sy Sx S y, (9) 0 for Sx Sy = 0

10 6 Zbgnew Bnderman, Bolesław Borkowsk, Wesław Szczesny Tversky ndex [Tversky 957] S S x y for Sx Sy > 0 xy Sx Sy + α S x \Sy + β S y \Sx, T = αβ 0. (0) 0 for Sx Sy = 0 Let us note that f n the above formula the coeffcents fulfll α=β= then we get Tanmoto s formula and f α=β= then we get Dce s formula. Here and n 2 the sequel we shall assume that α=β=. 4 Note that the defned above measures of smlarty, take a value between [0, ], are dependent on the orderng of features n case once represents the object by a radar chart. Another smple way of vsualzng the structure x = ( x, x2,..., xn ) s a bar graph, n whch each coordnate s represented as a rectangle of wdth and heght x (for =,,n). The area of such graph s equal to and one of the most popular ndcators of smlarty of two structures x = ( x, x2,..., xn ) and y = ( y, y2,..., y n ) s defned as [Malna 2004]: n Wxy : = mn( x, y), () = It s clear that ts value s ndependent of the orderng of features and, n the case of such graphcal representaton of structure, takes a value dentcal to the values of coeffcents defned n (6) and (8). In every stuaton when the ndcator of smlarty of two structures that uses a graphcal nterpretaton s not nvarant under the permutaton of coordnates, we may modfy ts defnton, n a way shown above (see formula (5)). Thus, to defne a measure of conformablty that would be ndependent of the orderng of features, let us denote by p j the j-th permutaton of numbers,2,,n. Naturally, each permutaton of coordnates of vectors x and y corresponds to one permutaton p j. Let vectors x j, y j denote j-th permutaton of coordnates of vectors x and y, respectvely. Assume that x =x, y =y, for each j-th permutaton x j, y j of coordnates of vectors x and y corresponds a coeffcent of conformablty of structures j c, = cxy, ( 2) j j QP

11 An applcaton of radar charts 7 where naturally c QP, = c xy, and the cosne smlarty c xy s defned as n formula (6). Wth regard to the above, let us assume the followng defntons of three dfferent measures of conformablty for objects Q and P M M j QP, = x,y = QP, j n! m m j QP, = x,y = QP, j n! n! s s j QP, = x,y = QP, n! j= c c max c, c c mn c, ( 3) c c c. In a smlar manner we can defne other coeffcents M m s M m s M m s M m s J xy,j xy,j xy;d xy,d xy, D xy ; Txy, Txy, T xy;t xy,txytxy. In order to demonstrate the presented above method of comparng structures, let us consder a smple example. Example. Let Q= x=,, 0, R= y =,,. Let us assume the followng denotatons: x: = x4 : = x, x2 : = x5: =,0,, x3: = x6 : = 0,,, y: = y2 : = y3: = y4 : = y5: = y6 = y. Thus we have: 2π 3 2π 3 Sx = sn =, S 3 sn, y = = π Sx S sn,, y = = S x S y = Rxy = / =, dla =,2,...,

12 8 Zbgnew Bnderman, Bolesław Borkowsk, Wesław Szczesny M m s 2 So Rx,y = Rx,y = Rx,y = = ~, M s m, where coeffcents RQP,, RQP,, R QP, are defned as n formulae (5). It can be easly verfed that coeffcents of conformablty of structures: cosne (formula (3)), Jaccard, Dce s are equal to: M m s M m s M m s c = c = c = 0, 385; J = J = J = 0, 236; D = D = D = 0, 38. x,y x,y x,y x,y x,y x,y x,y x,y x,y Note that M M M c = 0620, ; J = 0486, ; D = 067,. x,y x,y x,y It s also noteworthy that n ths case the coeffcent of conformablty of structures 2 (defned by formula ()) s W xy = =. The value of the coeffcent defne by formula (7) or (9) that uses an nterpretaton of the structure as a bar graph s equal to 0,5 =0,6(6)/,3(3). The above example shows that measures of smlarty of two objects calculated by dfferent methods (e.g. a method that ncludes the manner of the graphcal representaton of the structure or a method of normalzng, whch, when appled, causes the measure of the area of the unon of faces to take a value between [0, ]), can be sgnfcantly dfferent. A sngle measure of smlarty of objects can be far from optmal n the understandng of a gven expert. Furthermore, experts can dsagree on the meanngs of ndvdual measures. Thus t s safer to use, n the analyss of structures, a measure that s, for example, an average of several dfferent measures of smlarty [see: Breman 994]. EMPIRICAL RESULTS In order to verfy the approach descrbed n the prevous secton, we present an evaluaton of the sze of changes n demographcal structures of European countres between the years 999 and 2000, usng the dscussed above coeffcents. The followng Tables and 2 contan values of ndcators evaluatng the change of demographcal structures for 27 countres between years 999 and 200; wth an ndcaton what poston they occuped n the rankng of values of ndvdual measures as well as two parttons of countres nto 4 groups (columns C and C2). The partton s made based on the values of ndcator M (arthmetc mean of values of ndcators R, C, J, D and T) and ndcator W, whle the thresholds were defned as: A-d, A, A+d, where A denotes an average and d standard devaton.

13 An applcaton of radar charts 9 Table. Values and rankngs of ndcators evaluatng the smlarty of demographcal structures of 27 European countres n the years 999 and 200. Indcators are defned on the grounds of formulas: R - (3), C - (6), J - (7), D - (8), T-(0), M=(R+C+J+D+T)/5, W - () for the followng orderng of age ranges: 0-4, 5-24, 25-49, 50-64, 65-79, >80. The last two columns contan nformaton about the partton nto 4 groups, accordng to values of ndcators M and W, respectvely. No. country R C J D T M W R C J D T M W C C2 Austra 0,9676 0,9536 0,90 0,9534 0,9762 0,9524 0, Belgum 0,966 0,9427 0,893 0,9425 0,9704 0,947 0, Bulgara 0,9455 0,9038 0,8243 0,9037 0,9494 0,9054 0, Cyprus 0,939 0,8967 0,820 0,8963 0,9453 0,8964 0, Czech Republc 0,9347 0,8776 0,788 0,8776 0,9348 0,883 0, Denmark 0,978 0,9606 0,9242 0,9606 0,9799 0,9607 0, Estona 0,9634 0,9438 0,8933 0,9437 0,970 0,9430 0, Fnland 0,9469 0,923 0,8385 0,92 0,9540 0,928 0, France 0,9504 0,980 0,8482 0,978 0,9572 0,983 0, Germany 0,9624 0,9340 0,8762 0,9340 0,9659 0,9345 0, Greece 0,9364 0,8900 0,806 0,8899 0,947 0,899 0, Hungary 0,944 0,8950 0,800 0,8950 0,9446 0,8978 0, Ireland 0,9225 0,8754 0,7779 0,875 0,9334 0,8769 0, Italy 0,9620 0,9346 0,877 0,9345 0,9662 0,9349 0, Latva 0,9587 0,9386 0,8839 0,9384 0,9682 0,9375 0, Lthuana 0,945 0,9239 0,8577 0,9234 0,9602 0,9220 0, Luxembourg 0,976 0,9650 0,938 0,9647 0,9820 0,9630 0, Malta 0,9263 0,8809 0,7867 0,8806 0,9365 0,8822 0, Netherlands 0,953 0,9276 0,8646 0,9274 0,9623 0,9270 0, Poland 0,906 0,8464 0,733 0,8460 0,966 0,8497 0, Portugal 0,9357 0,8873 0,7973 0,8872 0,9402 0,8895 0, Romana 0,9346 0,8820 0,7888 0,8820 0,9373 0,8849 0, Slovaka 0,93 0,853 0,7435 0,8529 0,9206 0,8566 0, Slovena 0,9288 0,8667 0,7647 0,8667 0,9286 0,87 0, Span 0,9299 0,8860 0,7949 0,8857 0,9394 0,8872 0, Sweden 0,9689 0,9594 0,925 0,9592 0,9792 0,9576 0, Unted Kngdom 0,9700 0,9625 0,9272 0,9622 0,9808 0,9605 0, Source: own work Note that each of the frst 5 ndcators presented n Table, has an dentcal geometrcal nterpretaton of smlarty of structures, an ntersecton of two hexagons that represent those structures. They dffer only by the method used to normalze that area, so that the value of the ndcator of smlarty s between [0, ]. That s why all the ndcators, wth the excepton of ndcator R, they gve the same orderng of European countres, accordng to the smlarty of structures for the years 999 and 200. Small dfferences are vsble only n the case of ndcator R. The results do not change f we modfy the ndcator so that ts value s ndependent of the orderng of coordnates of the vector representng the structure (see. Table 2). On the other hand, dfferences between the orderng by the value of ndcator W (based on a dfferent vsualzaton of structures that the rest), and the

14 0 Zbgnew Bnderman, Bolesław Borkowsk, Wesław Szczesny orderng by the value of ndcator M are notceable. Even more so n the last two columns of Table 2, whch represent the partton of countres nto 4 groups, accordng to the smlarty of structures for years 999 and 200. In case of Span, we can observe a substantal dfference n the assgnment to a group dependng on the used ndcator. Table 2. Descrpton s smlar to that of Table. The calculatons of ndvdual ndcators where performed based on the frst formulas (maxmum) from (5), (3) and analogous modfcatons freeng the value of an ndcator from the orderng of coordnates of a vector descrbng a gven structure. No. country R M C M J M D M T M M M W R M C M J M D M T M M M W C C2 Austra 0,9737 0,9572 0,977 0,957 0,978 0,9567 0, Belgum 0,967 0,9427 0,893 0,9425 0,9704 0,9428 0, Bulgara 0,9658 0,9360 0,8796 0,9360 0,9669 0,9369 0, Cyprus 0,9592 0,9239 0,8575 0,9233 0,960 0,9248 0, Czech Republc 0,9582 0,926 0,8545 0,925 0,9592 0,9230 0, Denmark 0,982 0,9658 0,9338 0,9658 0,9826 0,9658 0, Estona 0,9702 0,952 0,9066 0,950 0,9749 0,9508 0, Fnland 0,9605 0,9294 0,8673 0,9290 0,9632 0,9299 0, France 0,9590 0,9234 0,8577 0,9234 0,9602 0,9247 0, Germany 0,9668 0,9386 0,8842 0,9385 0,9683 0,9393 0, Greece 0,9629 0,936 0,878 0,935 0,9646 0,9325 0, Hungary 0,9635 0,938 0,8722 0,937 0,9647 0,9328 0, Ireland 0,9665 0,9429 0,898 0,9428 0,9705 0,9429 0, Italy 0,9804 0,9669 0,9357 0,9668 0,983 0,9666 0, Latva 0,9632 0,946 0,8890 0,942 0,9697 0,940 0, Lthuana 0,9527 0,9254 0,8598 0,9246 0,9608 0,9247 0, Luxembourg 0,979 0,9650 0,938 0,9647 0,9820 0,9645 0, Malta 0,949 0,9097 0,8340 0,9095 0,9526 0,90 0, Netherlands 0,9653 0,9379 0,883 0,9379 0,9680 0,9384 0, Poland 0,9468 0,903 0,8233 0,903 0,949 0,905 0, Portugal 0,9680 0,9376 0,8826 0,9376 0,9678 0,9387 0, Romana 0,9548 0,972 0,8470 0,97 0,9568 0,986 0, Slovaka 0,9544 0,98 0,8378 0,98 0,9538 0,939 0, Slovena 0,9482 0,95 0,846 0,940 0,955 0,948 0, Span 0,9705 0,9425 0,893 0,9425 0,9704 0,9434 0, Sweden 0,984 0,9649 0,9322 0,9649 0,9822 0,965 0, Unted Kngdom 0,9790 0,9634 0,9293 0,9634 0,983 0,9633 0, Source: own work Tables and 2 show that the greatest stablty of the demographcal structure between 999 and 200 was possessed by: Austra, Denmark, Luxembourg, Sweden and Unted Kngdom. On the other hand, the greatest changes were observed n: Cyprus, Malta, Poland, Slovaka and Slovena. The greatest change occurred n Poland, and the smallest one n Luxembourg.

15 An applcaton of radar charts SUMMARY Means for defnng the values of ndcators of smlarty that use geometrcal nterpretatons n the form of a value of an area and are descrbed n ths work can also be used n other geometrcal ways of studyng the smlarty of structures as well as objects. These ways are an example of applyng geometrcal methods that are ntroduced by the authors usng radar charts [Bnderman, Borkowsk, Szczesny 2008, 200]. The emprcal analyss shows that when structures are not subject to large changes then the values of ndvdual ndcators, based on the same geometrcal nterpretaton, they order the structures smlarly. However, f we change the way of vsualzng the smlarty (the geometrcal nterpretaton) then we see changes n orderng. That s the reason why t s advsable to use several dfferent ndcators that use dfferent means of vsualzaton. Furthermore, t s worth notng that by usng geometrcal nterpretaton as a bass to construct an ndcator of smlarty we can obtan an ndcator that s very senstve to changes n the orderng of coordnates of a vector that numercally represents a gven structure. In practce there may be stuatons n whch a researcher desres such qualty n an ndcator so t may vsbly hghlght even small dfferences between structures, but for a gven orderng of ther components. However, one needs to remember that methods of constructng ndcators of smlarty that use a geometrcal nterpretaton are often appled manly because of the ease of vsualzaton of multdmensonal data. Then an unseasoned researcher may msuse them. It must be hghlghted that ndcators based solely on those llustratons do not satsfy often posed n the lterature on ths subject the basc requrement of stablty of the used method [see Jackson 970], that means the ndependence of the orderng of features. Technques presented by the authors show how a defnton of an ndcator must be modfed (the method of measurement) to remove ths flaw. Technques that were ponted out may seem numercally complex; nevertheless, n the age of computers that problem became nsgnfcant. On the other hand, ths smple and stable emprcal example shows that by applyng modfcatons, that s makng the measurement of smlarty ndependent of the orderng of ndvdual components of the structure, we obtan dfferent results (see Tables and 2, e.g., Span). The measurement of smlarty of structures based on geometrcal nterpretaton becomes even more complcated when a researcher s nterested n changes that occurred n a gven structures durng the whole studed perod and not only between the begnnng and the end of the sample. Further works on ths subject can be found n the work Bnderman and Szczesny 20.

16 2 Zbgnew Bnderman, Bolesław Borkowsk, Wesław Szczesny REFERENCES Bnderman Z., Borkowsk B., Szczesny W. (2008) O pewnej metodze porządkowana obektów na przykładze regonalnego zróżncowana rolnctwa, Metody loścowe w badanach ekonomcznych, IX, wyd. SGGW, Bnderman Z., Borkowsk B., Szczesny W (2009) O pewnych metodach porządkowych w analze polskego rolnctwa wykorzystujących funkcje użytecznośc, Rocznk Nauk Rolnczych PAN, Sera G, Ekonomka Rolnctwa, T. 96, z. 2, Bnderman Z., Borkowsk B., Szczesny W. (2009a) Tendences n changes of regonal dfferentaton of farms structure and area Quanttatve methods n regonal and sectored analyss / sc. ed. D. Wtkowska D., Łatuszyńska M., U.S., Szczecn, Bnderman Z., Borkowsk B., Szczesny W. (200) Radar measures of structures conformablty, Quanttatve methods n economy, v. XI, No., Bnderman Z., Borkowsk B., Szczesny W. (200a) The tendences n regonal dfferentaton changes of agrcultural producton structure n Poland, Quanttatve methods n regonal and sectored analyss, U.S., Szczecn, Bnderman Z., Borkowsk B., Szczesny W. (200b) Wykorzystane metod geometrycznych do analzy regonalnego zróżncowana kultury na ws, Sera T. XII, z. 5, Bnderman, Z., Borkowsk B., Szczesny W. (200c) Regonalne zróżncowane turystyk w Polsce w latach , Oeconoma 9 (3) Bnderman Z., Borkowsk B., Szczesny W. (200d) Regonalnego zróżncowane kultury mędzy wsą a mastem w latach , Mędzy dawnym a nowym na szlakach humanzmu, wyd. SGGW, Bnderman Z., Szczesny W. (2009) Arrange methods of tradesmen of software wth a help of graphc representatons Computer algebra systems n teachng and research, Sedlce, Wyd. WSFZ, 7-3. Bnderman Z., Szczesny W., (20) Comparatve Analyss of Computer Technques for Vsualzaton Multdmensonal Data, Computer algebra systems n teachng and research, Sedlce, wyd. Collegum Mazova, Bnderman Z. (2009) Syntetyczne mernk elastycznośc przedsęborstw, Kwartalnk Prace Materały Wydzału Zarządzana Unwersytetu Gdańskego, nr 4/2, Bnderman, Z. (2009a) Ocena regonalnego zróżncowana kultury turystyk w Polsce w 2007 roku Rocznk Wydzału Nauk Humanstycznych, SGGW, T XII, Borkowsk B., Szczesny W. (2002) Metody taksonomczne w badanach przestrzennego zróżncowana rolnctwa. Warszawa, RNR, Sera G, T 89, z Breman L. (994) Baggng predctors, Techncal Report 420, Departament of Statstcs, Unversty of Calforna, CA, USA, September 994. Cok A., Kowalczyk T., Pleszczyńska E., Szczesny W. (995) Algorthms of grade correspondence-cluster analyss. The Collected Papers on Theoretcal and Apled Computer Scence, 7, Deza E., Deza M.M. (2006) Dctonary of Dstances, Elsever. Dce Lee R. (945) Measures of the Amount of Ecologc Assocaton Between Speces, Ecology, Ecologcal Socety of Amerca, Vol. 26, No. 3,

17 An applcaton of radar charts 3 Duda R. O., Hart P. E., Stork D. G. (2000) Pattern Classfcaton. John Wley & Sons, Inc., 2nd ed. Gordon, A.D. (999) Classfcaton, 2nd edton, London-New York. Hubalek Z. (982) Coeffcents of Assocaton and Smlarty, Based on Bnary (Presence- Absence) Data: An Evaluaton, Bologcal Revews, Vol.57-4, Jaccard P. (90) Étude comparatve de la dstrbuton florale dans une porton des Alpes et des Jura. Bulletn del la Socété Vaudose des Scences Naturelles 37, Jaccard, P. (902) Los de dstrbuton florale dans la zone alpne. Bull. Soc. Vaud. Sc. Nat. 38, Jaccard, P. (908) Nouvel les recherches sur la dstrbuton floral e. Bull. Soc. Vaud. Sc. Nat. 44, Jackson D. M. (970) The stablty of classfcatons of bnary attrbute data, Techncal Report 70-65, Cornell Unversty, -3. Kukuła K. (2000) Metoda untaryzacj zerowej, PWN, Warszawa. Kukuła K. (red) (200) Statystyczne studum struktury agrarnej w Polsce, PWN, Warszawa. Legendre P., Legendre L. (998) Numercal Ecology, Second Englsh Edton Ed., Elsever. Malna A. (2004) Welowymarowa analza przestrzennego zróżncowana struktury gospodark Polsk według województw. AE, Sera Monografe nr 62, Kraków. Monev V. (2004) Introducton to Smlarty Searchng n Chemstry, MATCH Commun. Math. Comput. Chem. 5, Mostowsk A., Stark M.: Elementy algebry wyższej, PWN, Warszawa. Szczesny W. (2002) Grade correspondence analyss appled to contngency tables and questonnare data, Intellgent Data Analyss, vol. 6, 7-5. Tan P., Stenbach M., Kumar V. (2006) Introducton to Data Mnng, Pearson Educaton, Inc. Tanmoto, T.T. (957) IBM Internal Report 7th Nov. 957 Tanmoto T.T. (959) An Elementary Mathematcal Theory of Classfcaton and Predcton, IBM Program IBCFL. Tversky A. (957) Features of smlarty. Psychologcal Revew, 84(4) Warrens M. J. (2008) On Assocaton Coeffcents for 2 2 Tables and Propertes that do not depend on the Margnal Dstrbutons, Psychometrka Vol. 73, n. 4,

18 4 Zbgnew Bnderman, Bolesław Borkowsk, Wesław Szczesny

19 QUANTITATIVE METHODS IN ECONOMICS Vol. XII, No., 20, pp SOME REMARKS ON APLICATIONS OF ALGEBRAIC ANALYSIS TO ECONOMICS Zbgnew Bnderman, Marek Werzbck Department of Econometrcs and Statstcs Warsaw Unversty of Lfe Scences SGGW e-mals: zbgnew_bnderman@sggw.pl; marek_werzbck@sggw.pl Abstract: In ths paper, the author contnues the nvestgatons started n hs earler work [Bnderman 2009]. Here, problems of lnear equaton Dx=y wth the dfference operator D are studed. The work s an ntroducton to applcatons of the theory of rght nvertble operators to economcs. As an example, quotatons of KGHM on Warsaw Stock Exchange are consdered. Key words: algebrac analyss, rght nvertble operator, dfference operator, quotatons of Stock Exchange, Jacoban matrx In memory of Professor Krystyna Twardowska INTRODUCTION In mathematcs the term Algebrac Analyss s used n two completely dfferent senses [cf. Przeworska - Rolewcz 2000]. Here, meanng of Algebrac Analyss s closely connected wth theory of rght nvertble operators [cf. Przeworska - Rolewcz 988]. In the earler work of the author [Bnderman 2009] a new defnton of elastcty operators n algebras wth rght nvertble operators was proposed. The defnton uses logarthmc mappngs of algebrac analyss [cf. Przeworska-Rolewcz 998]. The obtaned results were appled to economcs n order to fnd a functon f elastcty of ths functon s gven. Here, possbltes of applcatons of algebrac analyss to economcs, on the smple example of the dfference operator D and the lnear equaton Dx=y are presented. The paper s an ntroducton n ths range.

20 6 Zbgnew Bnderman, Marek Werzbck Throughout ths work wll denote ether the real feld,, or the complex feld, Let X and Y be a lnear space over. The set of all lnear operators domans contaned n X and ranges contaned n Y wll be denoted by L(X,Y). We shall wrte: L0(X,Y): = { A L(X,Y):domA = X }, L(X): = L(X,X), L0(X) : = L0(X, X), ker A :{ x dom A :Ax = 0} for A L(X, Y). Followng D. Przeworska - Rolewcz [c.f. Przeworska - Rolewcz 988], an operator D L(X) s sad to be rght nvertble f there s an operator R L L 0 (X) such that RX dom D and DR=I. The operator R s called a rght nverse of D. We shall consder n L(X) the followng sets: the set R(X) of all rght nvertble operators belongng to L(X) ; the set R D := { R L 0 (X) : DRx = x for all x X }; the set F D := { F L 0 (X) : F 2 = F, FX = ker D and R R D : FR=0} of all ntal operators for a D R(X). We note, f D R(X), R R D and ker D {0 }, then the operator D s rght nvertble, but not nvertble. We have DRx= x for all x X and x dom D: RDx x. Here, the nvertblty of an operator A L (X) means that the equaton Ax=y has the unque soluton for every y X. If D R(X) and 0 z ker D and x s a soluton of the equaton DX=y then the element x +z s also the soluton of ths equaton. If F s an ntal operator for D correspondng to R then Fx =x RDx=(I-RD)x for x dom D and Fz=z for z ker D. () We note, a dfferent approach to the defnton of rght nvertble lnear operators s presented n the work [Bnderman 2009]. In the sequel we shall assume that D R(X), R R D, F F D s an ntal operator for D correspondng to R and dm ker D>0,.e. D s rght nvertble but not nvertble. We observe, that f we know one rght nverse of D then the sets [c.f. Przeworska - Rolewcz 988] R D = {R + FA: A L 0 (X)}; (2) F D = {F(I-AD): A L 0 (X)}. (3) We shall need the two followng theorems [c.f. Przeworska - Rolewcz 988]. Theorem. The general soluton of the equaton s gven by the formula Dx = y, y X, (4)

21 Some remarks on applcatons of algebrac analyss 7 x=z + Wy, (5) where z ker D s arbtrary and W R D s arbtrarly fxed. Theorem 2. The ntal value problem Dx = y, y X, (4) Fx= z 0, z 0 ker D, (6) has the unque soluton of the form x=z 0 +Ry, (7) where R R D s the rght nverse of operator D, correspondng to F. We consder the several examples of operators whch are often present n economcs. EXAMPLE. [cf. Przeworska Rolewcz 988, Bttner 974, Levy, Lessman 959, Gelfond 957]. We suppose that X s the set of all sequences x={x n }, where x n R, n N={,2, } wth addton and multplcaton by scalars defned as follows: f x={x n }, y={y n }, λ R then x+y=={x n +y n }, λx={λx n }. Defne the dfference operators by the equaltes: Dx={x n+ x n }, x={x n }, Rx={0, x, x +x 2, x +x 2 +x 3, }. For x={x n } X we have: DRx = D{0, x, x +x 2, x +x 2 +x 3, }={x -0, x +x 2 -x, x +x 2 +x 3 x 2 -x, }= ={x, x 2, x 3, }=x. RDx = R{x 2 x, x 3 x 2, x 4 x 3, }={0, x 2 x, x 2 x +x 3 x 2, x 2 x +x 3 x 2, x 2 x +x 3 x 2 +x 4 x 3, }= x-x e x for x 0, where e:={,,,, }. The above shows that D s not nvertble. The kernel of the operator D has the form: ker D = {z={z n }: z n =c, n N, c R}. By Theorem, the general soluton of the equaton (4) s of the form x={x n }=={c, c+x, c+x +x 2, c+x +x 2 +x 3, }, (8) where c s arbtrarly fxed. We observe that the ntal operator F for D correspondng to R s defned by the formula: Fx=(I-RD)x = x (x x e)= x e ker D, x={x n } X.

22 8 Zbgnew Bnderman, Marek Werzbck EXAMPLE 2. [cf. Przeworska-Rolewcz 988]. We denote by X=C[a,b] the set of all real-valued functon defned and contnuous on a closed nterval [a,b]. The set X s a lnear space over the feld of real numbers f the addton and multplcaton by a number are defned as follows: (x+y)(t)=x(t)+y(t); (αx)(t)= αx(t) for x,y X, t [a,b], α. Smlarly propertes has the set C [a,b] X of all real-valued functons defned on a closed nterval [a,b] and havng contnuous dervatve n (a,b). Suppose that we are gven a pont t 0 [a,b] and c s an arbtrary fxed real number. We defne operators as follows: (Dx)(t):= x (t) for x C [a,b] X; t [a,b], (Rx)(t)= x( τ ) t t dτ for x X; t [a,b]. 0 The defnton of D and R mples that R R D and (Fx)(t)=[(I-RD)x](t) =x(t 0 ) on dom D= C [a,b]. EXAMPLE 3. [cf. Bnderman 992, 993, 2000] Suppose that X s defned as n Example 2, where 0 (a,b). Let C0 [ a,b] Xdenotes the set of all realvalued functon defned on a closed nterval [a,b] and havng contnuous dervatve n the pont 0. We defne operators as follows: x(t) x(0) for t [a,b] t Dx (t), x C0 a,b. x'(0) for t= 0 ( ) = [ ] (Rx)(t)=tx(t), x X; t [a, b] The operator D s called a Pommez operator or a backward shft operator [Dmovsk 990, Dmovsk 2005, Douglas, Shapro, Shelds 970, Fage, Nagnbda 987, Lnchuk 988].The defnton of D, R mples that R R D and (Fx)(t) =[(I-RD)x](t) =x(0). EXAMPLE 4. [cf. Bnderman 2009]. In smlar way as n Example 2 we denote by X=C[a,b], where a>0, the set of all real-valued functon defned and contnuous on

23 Some remarks on applcatons of algebrac analyss 9 a closed nterval [a,b]. Suppose that we are gven a pont t 0 [a,b]. We defne the operators D as follows: (Dx)(t):=tx (t) for x C [a,b] X; t [a,b]. ( ) t x τ (Rx)(t)= dτ for x X; t [a,b]. τ t 0 The operator R s well-defned for all contnuous functons. The defnton of R mples that RX dom D= C [a,b] and ( τ ) x() t d t x (DRx)(t)=D(Rx(t))= t = t =x( ), dt dτ t τ t0 t The operator D s rght nvertble but not nvertble snce t t, t0 t0 for all x X ; t [a,b]. (Dx)( τ) τx ( τ) (RDx)(t)= d τ= d τ = x(t)-x(t 0) for x dom D. τ τ In ths case, the operator defned as follows (Fx)(t):= x(t)-(rdx)(t)=((i-rd)x)(t)=x(t 0 ) for x C [a,b]; t [a,b] s an ntal operator of D correspondng to the rght nverse R of D. We note, n the work of the author [Bnderman 2009] the operator D was used to construct an operator of elastcty. EXAMPLE 5. [cf. Bnderman 2000]. Suppose that X s defned as n Example 2. We defne the famly of operators D h as follows: and xt () xh ( ) h for t h, t h D hx ( t): = h ( a, b), x X, hx '( h) for t = h, ( ) t h R hx ( t): =,0 h ( a, b). h ( )

24 20 Zbgnew Bnderman, Marek Werzbck We can prove, the operators D h are rght nvertble, R h R D h and (F h x)(t) =[(I-R h D h )x](t) =x(h) for all 0 h (a,b), x X. DIFFERENCE EQUATIONS PROBLEM. Suppose that X, D, R s defned as n Example. Let a sequence x={x n } X denotes a pont seres or tme seres. We set the problem of fndng elements of x f we know the members: y, y 2,, y m of the sequence y={y n }, under the condton Dx=y. If x s gven then by the defnton of the operator D, only we receve: x 2 =y +x, x 3 =y 2 + y +x,.., xm+ x y m = +. Clearly, by Theorem 2 we obtan the same result x= x e +Ry = {x, x,, x, } + {0, y, y + y 2, y + y 2 + y 3, }. Let us consder the followng problem: fnd a soluton of the equaton Dx=y, (4) whch satsfes the lnear condton (a,x)=b, (9) where (, ax ): = ax, a,x,y X; b R. = In order to determne the members x,x 2,,x m of a sequence x, we assume that elements a,a 2,,a m ; y,y 2, y m- of the sequences a, y, respectvely are known and m a 0; aj = 0 for j > m. = We can prove that the problem s equvalent to the ntal value problem Dx = y, (4) wth the condton Fx= z 0, where b z0 : = m e ker D, e={,, }. (0) a It s easy to check that F s the ntal operator for D correspondng to the operator = =

25 Some remarks on applcatons of algebrac analyss 2 m m u = m aj y a = 2 j= Ry = R{ yn} = { un} : = =. () n un = u + yj for n > j = By the Theorem 2 we receve that x = z0 + Ry. Hence we receve the followng theorem. Theorem 3. The unque soluton of the problem (4), (9) s determned by the followng formula: m m b aj y n = 2 j= x =, xn = x+ yj for n= 2,3,..., m. (2) m j= a = We consder specal cases of the problem (4), (9). The above result mples the followng conclusons. Remark. [see also Przeworska-Rolewcz 988] Let the condton (9) has the form: ax p p = b, a p 0, p [, m ]. By formulas (0), () we obtan that the operator F (p) x:={b/a p }; x X s the ntal operator for D correspondng to the operator p x for n < p = n R(p) x= R(p) { x n} : = 0 for n = p, x X. n p xp+ for n > p = By Theorem 3 we obtan that the soluton of the consderng problem s determned by the formula b x= e+r (p) y. a Hence, the frst m members of the sequence x={x n } X s determned by the followng formula: p

26 22 Zbgnew Bnderman, Marek Werzbck b ap b xn = ap b + ap p = n n p = y y p+ for n =, 2..., p, for n = p, for n = p+, p+ 2,..., m Remark 2. [see also Przeworska - Rolewcz 988] Let the condton (9) has the k form: x = kx, k [2, m ], where x s gven. As t was ponted out n the k = k ntroducton, f D R(X), then there s the set of rght nverses, determned by Formula (2). Let x={x n } X, we defne the operator k ( k ) x, for n=, k = R kx: = y = { yn} =. n y + x for n>, = It s easy to check that R k R D and the operator k F kx: = x = x k e ker D, k = s the ntal operator for D correspondng to the operator R k. Hence, the frst k members of the sequence x={x n } X are determned by the followng formula: x n k x ( ) k k y k = = n x + y = for n =,. (3) for n = 2, 3,..., k, EXAMPLE 6. We consder the quotatons (n pln) of KGHM (KGHM Polska Medź S.A.) on Warsaw Stock Exchange for the perod August 20 2 August 20. Let x, x 2,, x 0 be unknown daly quotatons of KGHM n, 2, 3, 4, 5, 8, 9, 0,, 2 of August 20, respectvely. We can check that the average of the

27 Some remarks on applcatons of algebrac analyss 23 0 quotatons x0 : = x =70,7; the dfferences between the quotatons y =x + 0 = x, =,2,,9 are presented n Fgure. Fgure. Dfferences between the daly quotatons of KGHM on 2,, (n pln) dfference n zl Source: own calculatons Usng Theorem 3, we receve by Formula 2 the daly quotatons of KGHM n August 20, whch s presented n the last row of Table. Table. Daly quotatons of KGHM on, 2,, (n pln) y x Source: own calculatons Remark 3. We consder the equaton (4) Dx=y wth the condton ϕ ( x ) = ϕ ( x, x2,..., x m ) = 0, (4) m where the dfferentable mappng ϕ:, m and y, y2,..., ym are gven. It s easy to observe that the consdered problem s equvalent to the followng system of equatons

28 24 Zbgnew Bnderman, Marek Werzbck ϕ ( x, x2,..., xm ) = 0 x2 x = y. (5) xm xm = ym The Jacoban matrx of the above system s of the form ϕ ϕ ϕ x x2 x m 0 0. Jx (, x2,..., xm ) = We can check that the determnant of J - Jacoban s equal m ϕ J =. = x Under some addtonal condtons, we can prove [cf. Skorsk 969] the theorem. m If the Jacoban J does not vansh at a pont x0, then n a neghborhood of the pont x 0 there exsts an unque soluton of the system (5). We observe that n specal case, when m 2 ϕ( x ( ) 2, x2,..., xm ) mσ x xm = 0, where m σ> 0, xm = x are gven, then the problem has not a soluton or m j = has the solutons determned by the formula: c xn = c + where c s arbtrarly fxed. Indeed, the Jacoban j = n = y for n =,, (6) for n = 2, 3,..., m,

29 Some remarks on applcatons of algebrac analyss 25 ( ) m m m m ϕ 2( m ) J = = 2 x xm = x mxm 2( m ) x xm 0. = x = m m = = = m = On the other hand, the members y, y2,..., ym determne the standard devaton σ of the elements x, x2,..., x m. Indeed, by the formula (8) we have m m m x xm = x x = x x x + yj = yj, m = m m = 2 j= m = 2 j= m m j x xm = x + yj xj = yj yk. m m j= j= j= j= 2 k= The above shows that standard devaton σ s entrely determned by the known elements y, y 2,..., ym. We observe, the mnor of the (m-)th order of the element ϕ / x of the determnant J = By the Kronecker Cappell theorem we receve that consdered problem has not a soluton or the problem has nfnte number of solutons, determned by Formula (6). Remark 4. If for the consdered problem the members y, y 2,, y m- of a sequence y={y n } are known only. For example, next members of the y we can determne by the one of the smplest extrapolaton formulas [Ralston 965]: y = 2y y - lnear, n n n 2 yn = ( 5 yn + yn 3 4 yn 2) - quadratc, 2 where n=m, m+,,. Clearly, we can use the Newton nterpolaton formula [Ralston 965].

30 26 Zbgnew Bnderman, Marek Werzbck CONCLUSION It seems to the author that presented here consderatons wll permt us to use methods of Algebrac Analyss n much more complcated cases. The author ntends to show other applcatons to economcs wth another rght nvertble operators, equatons and condtons. Mathematcal theory of rght nvertble operators provdes good tools for solvng these problems. REFERENCES Agarwal R. P. (20000) Dfference Equatons and Inequaltes: Theory, Methods and Applcatons, Marcel Dekker, New York. Bnderman Z. (992) Functonal shfts nduced by rght nvertble operators, Math. Nachr. 57, Bnderman Z. (993) An unfed approach to shfts nduced by rght nvertble operators, Math. Nachr. 6, Bnderman Z. (994) On some functonal shfts nduced by operators complex dfferentaton, Opuscula Mathematca, Vol. 4, Bnderman Z. (2000)On rght nverses for functonal shfts, Demonstrato Mathematca, Vol. XXXIII, No 3, Bnderman Z. (2009) On elastcty operators and ther economcal applcatons, Polsh Journal an Envronment Study, Vol. 8, no. 5B, 2009, s Bttner R. (974) Rachunek operatorów w przestrzenach lnowych, PWN Warszawa. Dmovsk I., H. (990) Convolutonal Calculus, Kluwer Acad. Publshers, Dordrecht. Dmovsk I., H., Hrstov V., Z. (2005) Commutans of the Pommez operator, IJMMS, Hndaw Publshng Corporaton, Douglas R.G., Shapro H.S., Shelds A.L. (970) Cyclc vectors and nvarant spaces for backward shft operator, Ann. Inst. Fourer 20,, Fage M.K., Nagnbda N.I. (987) An equvalence problem of ordnary lnear dfferental operators, Nauka, Novosybrsk (n Russan) Gelfond A.O. (957) Calculus of fnte dfference, FzMatGz, Moskva (Russan) Levy H., Lessman F. (959) Fnte dfference equatons, Ptman LTD London. Lnchuk N. (988) Representaton of commutants of the Pommez operator and ther applcatons. Mat. Zametk 44, (n Russan) Przeworska-Rolewcz D. (988) Algebrac Analyss, D. Redel and PWN Polsh Scentfc Publshers, Dordrecht-Warszawa. Przeworska-Rolewcz D. (998) Logarthms and antlogarthms. An Algebrac Analyss Approach. Wth Appendx by Z. Bnderman, Kluwer Academc Publshers, Dordrecht. Przeworska-Rolewcz D. (2000), Two centures of the term Algebrac Analyss, Algebrac Analyss and related topcs, Banach Center Publcatons, V.53, 47-70, Warsaw. Ralston A. (965) A Frst Course n Numercal Analyss, McGraw-Hll Book Company. Skorsk R. (969) Advanced Calculus, Functon of several varables, Monografe Matematyczne, PWN Warszawa.

31 QUANTITATIVE METHODS IN ECONOMICS Vol. XII, No., 20, pp TRADING VOLUME AND VOLATILITY OF STOCK RETURNS: EVIDENCE FROM SOME EUROPEAN AND ASIAN STOCK MARKETS Mchaela Chocholatá Department of Operatons Research and Econometrcs Unversty of Economcs n Bratslava e-mal: chocholatam@yahoo.com Abstract: Ths paper analyses the relatonshp between the daly volatlty of stock returns and the tradng volume usng the TGARCH models for selected European and Asan stock markets. The leverage effect has been proved n all analysed cases. The logarthm of the tradng volume was ncluded nto the condtonal volatlty equaton as a proxy for nformaton arrval tme. Although n case of all analysed Asan stock returns the ncluson of the tradng volume led to the moderate declne of the condtonal volatlty persstence, the results n case of European stock returns were not so unambguous. Key words: volatlty, TGARCH model, tradng volume, stock returns INTRODUCTION The analyss of the stock returns volatlty has attracted the nterest of nvestors for a long tme. One of the characterstc features of stock returns s that ther volatlty changes over tme. Although ths feature has long been recognzed (see e.g. [Franses et al. 2000]), the poneerng work n the area of modellng volatlty was presented by Engle [Engle 982] who ntroduced the autoregressve condtonal heteroscedastcty model ARCH. The generalzed verson of ths model, the GARCH model, was frst publshed by Bollerslev [Bollerslev 986]. The man am of the ARCH and GARCH models s to capture the tme-varyng Ths paper s supported by the Grant Agency of Slovak Republc - VEGA, grant no. /0595/ "Analyss of Busness Cycles n the Economes of the Euro Area (wth Regards to the Specfcs of the Slovak Economy) Usng Econometrc and Optmzaton Methods"

32 28 Mchaela Chocholatá volatlty. In order to capture some other typcal features of fnancal tme seres such as asymmetrc effect (.e. the dfferent mpact of the postve and negatve shocks on the condtonal volatlty) or long memory (.e. varances generated by fractonally ntegrated processes) a large number of modfcatons of the standard ARCH and GARCH models has been developed and t s almost mpossble to menton all of them (see e.g. [Franses et al. 2000], [Poon et al. 2003], [Rachev et al. 2007]). To capture the asymmetrc behavour of the stock returns e.g. the Engle s [Engle 990] asymmetrc GARCH (AGARCH) model, Nelson s [Nelson 99] exponental GARCH (EGARCH) model or Zakoan s [Zakoan 994] threshold GARCH (TGARCH) model can be used. From the long memory models, the most popular and well known s the fractonal ntegrated GARCH (FIGARCH) model of Balle, Bollerslev and Mkkelsen [Balle et al. 996]. Though the ARCH/GARCH class models allow the volatlty shocks to persst over tme, they ddn t provde the economc explanaton for ths phenomenon. The paper [Lamoureux et al. 990] publshed n the Journal of Fnance offers the explanaton for volatlty persstence. The authors proved that the daly tradng volume, used as a proxy for nformaton flow, has a sgnfcant explanatory power regardng the varance of daly returns. For a sample of 20 US actvely traded stocks they found out that the GARCH effects dsappeared when the tradng volume was ncluded nto the condtonal varance equaton. The number of studes documentng the relatonshp between the stock returns and tradng volume s constantly growng. The survey of some emprcal studes dealng wth ths relatonshp can be found e.g. n [Ghysels et al. 2000], [Grard et al. 2007], [Gursoy et al. 2008], [Poon et al. 2003]. The above mentoned approach presented n [Lamoureux et al. 990] has been appled n varous studes to both ndvdual stocks (stock-level analyss) and stock market ndces (market-level analyss). Snce the conclusons of the studes applyng the approach of Lamoureux and Lastrapes on ndvdual stocks are mostly n concdence wth those presented n [Lamoureux et al. 990], the results of the market-level analyss are not so unambguous (see e.g. [Grard et al. 2007], [Sharma et al. 996]). There are also some papers whch proved that the ncluson of the tradng volume n condtonal varance equaton elmnates the ARCH effect for both the ndvdual stocks and the stock ndex (see e.g. [Myakosh 2002]). The analyses n the studes [Grard et al. 2007] and [Gursoy et al. 2008] were done also for the decomposed total volume (nto ts predctable and unpredctable components) to examne the role of dfferng tradng systems on the relatonshp between the condtonal volatlty and the tradng volume. The am of ths paper s to analyse the relatonshp between the tradng volume and the daly volatlty of eght European and fve Asan stock returns data (.e. market-level analyss) usng the TGARCH models and applyng the approach TGARCH model s equvalent to the GJR GARCH model ndependently presented by Glosten, Jagannathan and Runkle [Glosten et al. 993].